Dynamics of Force Dipoles in Curved Biological Membranes

TL;DR

This paper models the hydrodynamic interactions of active force dipoles in curved biological membranes, revealing effects of curvature on flow fields, defect formation, and dipole aggregation behavior.

Contribution

It introduces an analytic model for force dipoles on curved membranes, incorporating curvature effects and topology, which was not addressed in prior planar membrane studies.

Findings

Flow fields include a negative index defect due to topology.

Dipoles on a sphere follow geodesics, acting as 'curvature checkers'.

High curvature can disrupt dipole aggregation even with confinement.

Abstract

We construct a model to explore the hydrodynamic interactions of active inclusions in curved biological membranes. The curved membrane is modelled as a two dimensional layer of highly viscous fluid, surrounded by external solvents of different viscosities. The active inclusions are modelled as point force dipoles. The point dipole limit is taken along a geodesic of the curved geometry, incorporating the change in orientation of the forces due to curvature. We demonstrate this explicitly for the case of a spherical membrane, leading to an analytic solution for the flow generated by a single inclusion. We further show that the flow field features an additional defect of negative index, arising from the membrane topology, which is not present in the planar version of the model. We observe that a mutually perpendicular dipole pair moves along geodesics on the sphere and thus act as "curvature checkers", analogous to vortex dipoles. We finally explore the hydrodynamic interactions of a pair of inclusions in regimes of low and high curvature, as well as situations where the external fluid outside the membrane is confined. Our study suggests aggregation of dipoles in curved biological membranes of both low and high curvatures, under strong confinement. However, very high curvatures tend to destroy dipole aggregation, even under strong confinement.